Continuing to Payment will take you to apayment page

Purchasing
SparkNotes PLUS
for a group?

Get Annual Plans at a discount when you buy 2 or more!

Price

$24.99$18.74/subscription + tax

Subtotal $37.48 + tax

Save 25%
on 2-49 accounts

Save 30%
on 50-99 accounts

Want 100 or more?
Contact us
for a customized plan.

Continuing to Payment will take you to apayment page

Your Plan

Payment Details

Payment Details

Payment Summary

SparkNotes Plus

You'll be billed after your free trial ends.

7-Day Free Trial

Not Applicable

Renews October 2, 2023September 25, 2023

Discounts (applied to next billing)

DUE NOW

US $0.00

SNPLUSROCKS20 | 20%Discount

This is not a valid promo code.

Discount Code(one code per order)

SparkNotes PLUS
Annual Plan - Group Discount

Qty: 00

SubtotalUS $0,000.00

Discount (00% off)
-US $000.00

TaxUS $XX.XX

DUE NOWUS $1,049.58

SparkNotes Plus subscription is $4.99/month or $24.99/year as selected above. The free trial period is the first 7 days of your subscription. TO CANCEL YOUR SUBSCRIPTION AND AVOID BEING CHARGED, YOU MUST CANCEL BEFORE THE END OF THE FREE TRIAL PERIOD. You may cancel your subscription on your Subscription and Billing page or contact Customer Support at custserv@bn.com. Your subscription will continue automatically once the free trial period is over. Free trial is available to new customers only.

Choose Your Plan

Your Free Trial Starts Now!

For the next 7 days, you'll have access to awesome PLUS stuff like AP English test prep, No Fear Shakespeare translations and audio, a note-taking tool, personalized dashboard, & much more!

Thank You!

You’ve successfully purchased a group discount. Your group members can use the joining link below to redeem their group membership. You'll also receive an email with the link.

No URL

Copy

Members will be prompted to log in or create an account to redeem their group membership.

Thanks for creating a SparkNotes account! Continue to start your free trial.

Please wait while we process your payment

Your PLUS subscription has expired

We’d love to have you back! Renew your subscription to regain access to all of our exclusive, ad-free study tools.

One of the main uses of an xy-graph is to graph equations. If an equation has both an x and y variable, then it often has multiple solutions of the form (x, y). In fact, there are generally infinitely many solutions to an equation with two variables.

The solutions to an equation in two variables can be represented by a curve on an xy-graph; every point on the curve has coordinates which satisfy the equation. In fact, for linear equations (our only concern in this chapter), the curve representing the solutions to the equation will actually be a straight line.

Example. Here is the graph of 2y - x = 4:

If we pick any point on the line -- (2, 1), (- 4, 0), or (, 2), for example -- it will satisfy the equation 2y - x = 4. Try a few points; they need not have integer values.

Making Data Tables

One way to graph an equation is by use of a data table. A data table is a list of x-values and their corresponding y-values. To make a data table, draw two columns. Label one column x and the other column y. Then list the x-values -2, - 1, 0, 1, 2 in the x column:

Next, plug each value of x into the equation and solve for y. Insert these values of y into the table, under their corresponding x values. For this example, we will use the equation 2x - 4 = 3y: x = - 2: 2(- 2) - 4 = 3y. 3y = - 8. y = - 2 x = - 1: 2(- 1) - 4 = 3y. 3y = - 6. y = - 2 x = 0: 2(0) - 4 = 3y. 3y = - 4. y = - 1 x = 1: 2(1) - 4 = 3y. 3y = - 2. y = - x = 2: 2(2) - 4 = 3y. 3y = 0. y = 0 Thus, the data table looks like:

Making Graphs Using Data Tables

To make a graph using the data table, simply plot all the points and connect them with a straight line. Extend the line on both sides and add arrows. This is to show that the line continues infinitely, even after it can be seen on the graph. Here is our sample data table as a graph:

Note that the large dots on the line are unnecessary -- they are merely there to show the data points we plotted. To check, pick a data point that is on the line but not in the chart -- it should satisfy the equation.

Notice also that it is not necessary to make a huge data table to graph a linear equation effectively. There is only one line through any two points, so already if one plots three points from a data table the redundancy of the third point acts as a check of the calculations. Of course, for more general equations whose graph does not consist of a straight line, more points are necessary to get an idea of the appearance of the graph.